Answer to Question #145141 in Statistics and Probability for Shafeeque

Question

It has been observed that the mean breaking strength of a particular brand of thread is 9.72N with a standard deviation of 1.4N. When a sample of 30 pieces is taken it showed a mean breaking strength of 8.93N. Can it be concluded that at the 5% level the thread has become inferior?

Answer 1

The provided sample mean is 

$\bar X = 8.93$  and the known population standard deviation is σ=1.4, and the sample size is n = 30.

The following null and alternative hypotheses need to be tested:

$\\ H_0:μ≥9.72\\ Ha:μ<9.72$

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

The z-statistic is computed as follows:

$z=\dfrac{\overline{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}$

$z=\dfrac{8.93-9.72}{\frac{1.4}{\sqrt{30}}}$ = −3.091

Using Z table, are

The p-value is p = 0.001, P(Z<=-3.091)= 0.001,since p=0.001<0.05, it is concluded that the null hypothesis is rejected.

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 9.72, at the 0.05 significance level.